Difference sets are not multiplicatively closed

We prove, in particular, that for any finite set of real numbers A with |A/A| \ll |A| one has |A-A| > |A|^{5/3 - o(1)}. Also we show that |3A| > |A|^{2-o(1)} in the case.

In our paper we study multiplicative properties of difference sets $A-A$ for large sets $A \subseteq \mathbb{Z}/q\mathbb{Z}$ in the case of composite $q$. We obtain a quantitative version of a result of A. Fish about the structure of the product sets $(A-A)(A-A)$. Also, we show that the multiplicative covering number of any difference set is always small.

There exists an absolute constant $C$ with the following property. Let $A \subseteq \mathbb{F}_p$ be a set in the prime order finite field with $p$ elements. Suppose that $|A| > C p^{5/8}$. The set \[ (A \pm A)(A \pm A) = \{(a_1 \pm a_2)(a_3 \pm a_4) : a_1,a_2,a_3,a_4 \in A\} \] contains at least $p/2$ elements.

We compare the size of the difference set $A-A$ to that of the set $kA$ of $k$-fold sums. We show the existence of sets such that $|kA| < |A-A|^{a_k}$ with $a_k<1$.

There exists an absolute constant $\delta > 0$ such that for all $q$ and all subsets $A \subseteq \mathbb{F}_q$ of the finite field with $q$ elements, if $|A| > q^{2/3 - \delta}$, then \[ |(A-A)(A-A)| = |\{ (a -b) (c-d) : a,b,c,d \in A\}| > \frac{q}{2}. \] Any $\delta < 1/13,542$ suffices for sufficiently large $q$. This improves the condition $|A| > q^{2/3}$, due to Bennett, Hart, Iosevich, Pakianathan, and Rudnev, that is typical for such questions. Our proof is based on ...

Given a finite set $A\subset \mathbb{R}\backslash \{0\}$, define \begin{align*}&A\cdot A \ =\ \{a_i\cdot a_j\,|\, a_i,a_j\in A\},\\ &A/A \ =\ \{a_i/a_j\,|\,a_i,a_j\in A\},\\ &A + A \ =\ \{a_i + a_j\,|\, a_i,a_j\in A\},\\ &A - A \ =\ \{a_i - a_j\,|\,a_i,a_j\in A\}.\end{align*} The set $A$ is said to be MPTQ (more product than quotient) if $|A\cdot A|>|A/A|$ and MSTD (more sum than difference) if $|A + A|>|A - A|$. Since multiplication and addition are commutative while divisio...

We prove new results on additive properties of finite sets $A$ with small multiplicative doubling $|AA|\leq M|A|$ in the category of real/complex sets as well as multiplicative subgroups in the prime residue field. The improvements are based on new combinatorial lemmata, which may be of independent interest. Our main results are the inequality $$ |A-A|^3|AA|^5 \gtrsim |A|^{10}, $$ over the reals, "redistributing" the exponents in the textbook Elekes sum-product inequality a...

We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+\epsilon}$ with sufficiently small $\epsilon > 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \geq 2$. The result can be seen as a real analog of the conjecture of S\'ark\"ozy that multiplicative subgroups of finite fields of prime order are additively irreducible.

We consider the multiplicative structure of sets of the form AA+1, where where A is a large, finite set of real numbers. In particular, we show that the additively shifted product set, AA+1 must have a large part outside of any generalized geometric progression of comparable length. We prove an analogous result in finite fields as well.

This is a sequel to the paper arXiv:1312.6438 by the same authors. In this sequel, we quantitatively improve several of the main results of arXiv:1312.6438, and build on the methods therein. The main new results is that, for any finite set $A \subset \mathbb R$, there exists $a \in A$ such that $|A(A+a)| \gtrsim |A|^{\frac{3}{2}+\frac{1}{186}}$. We give improved bounds for the cardinalities of $A(A+A)$ and $A(A-A)$. Also, we prove that $|\{(a_1+a_2+a_3+a_4)^2+\log a_5 : a_i...